Lang before Spivak

I'm looking to begin an independent study of Calculus, having read Lang's Basic Mathematics. I would like to know if I should move straight on to Spivak's Calculus, or start with Lang's A First Course in Calculus? If it affects the answer in any way, I have had some rudimentary experience with calculus and proofs, but nothing truly concrete.

I strongly advice against starting with Spivak. The book is excellent and certainly something you should read, but not as a first encounter with calculus. Lang's first course in calculus is very suitable as introduction to calculus. It covers the topics intuitively, but not too rigorous. So I think that doing Lang first, and then Spivak or Apostol is a very good choice!

I took Calculus I this last semester, and I'm currently working through the Lang book over break to keep it fresh and to get some different perspective. It's a very well written book with a lot of clear proofs and relevant examples. I aced calc I, but there are still some aspects and ideas in the Lang book that were never covered in the course. I'm also using it to help get a little bit ahead on some of the main ideas of Calculus II. From what I've seen of it so far, I highly recommend the Lang book.

I've heard that the Spivak book is a lot more challenging, and not really practical as an introduction. I plan on picking it up sometime down the line though.

I'd also add that I agree regarding limits. The treatment of limits in Lang's book is fairly cursory. It definitely doesn't contain the rigorous treatment that I got in my calc I course.

That is completely true, but that's Lang's intent. He finds that a rigorous treatment of limits is not something to waste time on in a first calc course. People were doing calculus for hundreds of years before they made sense of limits, so it's not that a rigorous understanding of limits is essential to calculus. People might disagree with this of course, but I think he has a point. More rigorous books like Spivak or Apostol do treat limits the correct way, but they're more second course calc books or even analysis books.

That is completely true, but that's Lang's intent. He finds that a rigorous treatment of limits is not something to waste time on in a first calc course. People were doing calculus for hundreds of years before they made sense of limits, so it's not that a rigorous understanding of limits is essential to calculus. People might disagree with this of course, but I think he has a point. More rigorous books like Spivak or Apostol do treat limits the correct way, but they're more second course calc books or even analysis books.

Yes, this is quite true. We covered Epsilon-Delta in my calculus course, but never really did much with it. The first section of chapter one (the chapter on limits) was focused on the Epsilon-Delta definition, and it seemed rather confusing and unnecessary. It seems redundant to prove limits before you even really understand the concept of what a limit actually IS. That was one of the more abstract ideas out of the whole course. It doesn't seem very productive to start the course with it.

Regardless, I still feel like Lang could have incorporated a slightly more comprehensive treatment of limits without getting into Epsilon-Delta. The first part of the book seems kind of backwards in some ways. He covers the limit definition of a derivative before he even formally covers limits. That struck me as odd. Perhaps other courses are different, but it seems more natural to me to start a calculus course with limits, rather than jumping right into derivatives. That may just be due to the fact that I learned it in that order though.